The Annals of Probability

Applications of Stein’s method for concentration inequalities

Sourav Chatterjee and Partha S. Dey

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Abstract

Stein’s method for concentration inequalities was introduced to prove concentration of measure in problems involving complex dependencies such as random permutations and Gibbs measures. In this paper, we provide some extensions of the theory and three applications: (1) We obtain a concentration inequality for the magnetization in the Curie–Weiss model at critical temperature (where it obeys a nonstandard normalization and super-Gaussian concentration). (2) We derive exact large deviation asymptotics for the number of triangles in the Erdős–Rényi random graph G(n, p) when p ≥ 0.31. Similar results are derived also for general subgraph counts. (3) We obtain some interesting concentration inequalities for the Ising model on lattices that hold at all temperatures.

Article information

Source
Ann. Probab., Volume 38, Number 6 (2010), 2443-2485.

Dates
First available in Project Euclid: 24 September 2010

Permanent link to this document
https://projecteuclid.org/euclid.aop/1285334211

Digital Object Identifier
doi:10.1214/10-AOP542

Mathematical Reviews number (MathSciNet)
MR2683635

Zentralblatt MATH identifier
1203.60023

Subjects
Primary: 60E15: Inequalities; stochastic orderings 60F10: Large deviations
Secondary: 60C05: Combinatorial probability 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

Keywords
Stein’s method Gibbs measures concentration inequality Ising model Curie–Weiss model large deviation Erdős–Rényi random graph exponential random graph

Citation

Chatterjee, Sourav; Dey, Partha S. Applications of Stein’s method for concentration inequalities. Ann. Probab. 38 (2010), no. 6, 2443--2485. doi:10.1214/10-AOP542. https://projecteuclid.org/euclid.aop/1285334211


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